## Abstract

Let G⩽Sym(Ω) be a finite almost simple primitive permutation group with socle G_{0}. A subset of Ω is a base for G if its pointwise stabilizer is trivial; the base size of G, denoted b(G), is the minimal size of a base. We say that G is standard if G_{0}=A_{n} and Ω is an orbit of subsets or partitions of {1,…,n}, or if G_{0} is a classical group and Ω is an orbit of subspaces (or pairs of subspaces) of the natural module for G_{0}. The base size of a standard group can be arbitrarily large, in general, whereas the situation for non-standard groups is rather more restricted. Indeed, we have b(G)⩽7 for every non-standard group G, with equality if and only if G is the Mathieu group M_{24} in its natural action on 24 points. In this paper, we extend this result by classifying the non-standard groups with b(G)=6. The main tools include recent work on bases for actions of simple algebraic groups, together with probabilistic methods and improved fixed point ratio estimates for exceptional groups of Lie type.

Original language | English |
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Pages (from-to) | 38-74 |

Number of pages | 37 |

Journal | Journal of Algebra |

Volume | 516 |

Early online date | 11 Sep 2018 |

DOIs | |

Publication status | Published - 15 Dec 2018 |

## Keywords

- Primitive permutation groups
- base sizes
- simple groups